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In mathematics, continuous geometry is an analogue of complex projective geometry introduced by , where instead of the dimension of a subspace being in a discrete set 0, 1, ..., ''n'', it can be an element of the unit interval (). Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor. ==Definition== Menger and Birkhoff gave axioms for projective geometry in terms of the lattice of linear subspaces of projective space. Von Neumann's axioms for continuous geometry are a weakened form of these axioms. A continuous geometry is a lattice ''L'' with the following properties *''L'' is modular. *''L'' is complete. *The lattice operations ∧, ∨ satisfy a certain continuity property. where ''A'' is a directed set and if α<β then ''a''α <''a''β, and the same condition with ∧ and ∨ reversed. *Every element in ''L'' has a complement (not necessarily unique). A complement of an element ''a'' is an element ''b'' with ''a''∧''b''=0, ''a''∨''b''=1, where 0 and 1 are the minimal and maximal elements of ''L'' *''L'' is irreducible: this means that the only elements with unique complements are 0 and 1. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Continuous geometry」の詳細全文を読む スポンサード リンク
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